1981 AMC 12 Problem 22

Question

How many lines in a three-dimensional rectangular coordinate system pass through four distinct points of the form

(i,j,k),

where

i,

j,

and

k

are positive integers not exceeding four?

60

64

72

76

100

Four collinear lattice points in this

4

-by-

4

-by-

4

grid must advance by coordinate steps

0

or

\pm1

Count axis-parallel lines, face-diagonal lines, and space diagonals separately

There are

3\cdot4^2=48

axis-parallel lines. For face-diagonal directions, choose the pair of varying coordinates in

3

ways, choose one of

2

diagonal slopes, and fix the remaining coordinate in

4

ways, giving

3\cdot2\cdot4=24.

Finally, the cube has

4

space diagonals. Thus the total is

48+24+4=76.

Therefore, the correct answer is D.

Problem 22 in other years